Fig 1.
Food web interaction matrix and application of perfect matching.
a, White (gray) boxes indicate nonzero (zero) matrix elements, orange boxes are unity matrix elements for the primary producers; dark and light blue squares indicate a possible path chosen, allowing to be nonzero. Here, No = n1 + n3 + n5 = 13 and Ne = n2 + n4 + n6 = 12, and Eq 6 is fulfilled with Δ = 1. Inset: Schematic of a possible pairing for the chosen path. Note that the invariance property of
was used, yielding only n1 non-vanishing matrix elements in the lower right block (Details: SI). b, Perfect matching [25] applied to simple food webs where competitive exclusion rules out coexistence due to lack of niches (i) and where enough niches are available for coexistence (ii). (iii) and (iv) are two additional examples, where coexistence is ruled out by the assembly rules. In (iii), n1 = 2, n2 = n3 = 1. In (iv), n1 = n3 = n4 = 1, n2 = 2. In both, Δ ≡ No − Ne ∉ {0, 1}, see Eq 5.
Fig 2.
Species richness for different trophic levels.
a, Food web with two trophic levels only; a staircase of coexistence with balanced species richness at levels 1 and 2 [18]. b, Three trophic levels. The number of intermediate species must equal the total number of basal and predator species. Intermediate species dominate ecosystem biodiversity. c, Four trophic levels. n2 (n3) must at least match basal (predator) species richness n1 (n4), indicated by thin black lines in green and orange bar. Solid green (orange) bars show the minimal upper bound to species richness in trophic level one (two). Species richness n2 and n3 can increase even further by co-evolution of intermediate species (shaded region). Note the applicable assembly rules shown for the different cases.
Fig 3.
a, Consumer limited food web and its interaction matrix. The symbols “×” mark nonzero entries and circles a path through the matrix. Basic nutrient is shown as a gray hexagon, whereas species on subsequent trophic levels are shown with coloured circles. Small circles highlight limitations by consumers. Shaded ovals indicate possible pairing of species. b, Possible assembly of the food web in (a) with labels for No+Ne and No−Ne. We set all growth and coupling constants equal to unity, but consider fine-tuned decay coefficients α ≪ 1 (details: [38], in prep.). Note the transitions between biomass limiting states. Sizes of circles indicate approximate densities of species in the different states. Link shown in gray was set to zero in the numerical simulations in (c). c, Species added one-by-one as shown in (b) and granted small initial population density (10−4). After each addition the system is integrated until steady state is reached. In the plot, colors of curves denote species of similar colors in the panels of (b). Note the double-logarithmic axis-scaling in (c). Time in each panel is relative to the time of introduction of the new species.
Fig 4.
Food web and corresponding interaction matrix for several species with sharp trophic levels and a single generalist omnivore. We only show its paired link in the network plot, but indicate all its interactions in the matrix.
Fig 5.
Comparison to existing models and data analysis.
a, Bahia Falsa free-living food web, with blue and red matrix elements for predator respectively prey dependency. Trophic levels indicated by gray and black bars, whereas color coding along the left and upper edge labels chain length of free-living species (increase from red to blue shades). b, Niche model simulation of the Bahia Falsa free-living food web. c, Cascade model simulation of the Bahia Falsa free-living food web. d, As (a) but with parasites (“Par”) and with colors for free-living species carried over from (a). Remaining species are parasites. e, As (d) but additionally including symmetric concomitant links (“ParCon sym”). f, Lack of niches (d) for seven empirical food webs [31]. Labels mark the sub-webs of free-living species (“Free“), including also parasite links (”Par“), asymmetric (”ParCon asym“) and several symmetric concomitant links (”ParCon sym“). Dashed line connects averages in these categories. Rank deficiencies for free living (df), free-living and parasite species (df+p) as well as additional concomitant links (dc,sym) marked in (a), (d), and (e), respectively. (Analysis details and abbreviations: see Methods).
Fig 6.
Simulations of different food web matrices.
a, Barplots indicate distributions of node richness for each approximate trophic level in the seven empirical foodwebs and a generic foodweb derived by averaging the empirical node richnesses in each trophic level. b, Interaction matrix corresponding to the generic food web, containing 110 free-living and 47 parasite species (Details: Methods and Sec. S10 in S1 Text). c, Addition of parasites that form random links to any existing free-living species. d, Addition of parasites that are confined to consumer at a specific trophic level. e, Similar to (c) but with the restriction of parasites consuming only free-living species at levels 3 and 4. f, Similar to (e) but with additional parasite-parasite interactions (hyperparasitism), approximately 5 percent of parasite links are from parasite to parasite. Note the color coding along the edges of the matrices in (c)—(f), chosen as in Fig 5a, 5d and 5e. g, The lack of niches, i.e. rank deficiency, as a function of the number of links per parasite for each of the four cases described in (c)—(f).